A373277 -id:A373277 - cp's OEIS frontend

A110540 Invertible triangle: T(n,k) = number of k-ary Lyndon words of length n-k+1 with trace 1 modulo k.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 3, 2, 1, 0, 3, 6, 5, 2, 1, 0, 5, 16, 16, 8, 3, 1, 0, 9, 39, 51, 30, 12, 3, 1, 0, 16, 104, 170, 125, 54, 16, 4, 1, 0, 28, 270, 585, 516, 259, 84, 21, 4, 1, 0, 51, 729, 2048, 2232, 1296, 480, 128, 27, 5, 1, 0, 93, 1960, 7280, 9750, 6665, 2792, 819, 180, 33, 5, 1

Offset: 1

Paul Barry, Jul 25 2005

An invertible number triangle related to Lyndon words of trace 1.

			Rows begin
  1;
  0,  1;
  0,  1,   1;
  0,  1,   1,    1;
  0,  2,   3,    2,    1;
  0,  3,   6,    5,    2,    1;
  0,  5,  16,   16,    8,    3,   1;
  0,  9,  39,   51,   30,   12,   3,   1;
  0, 16, 104,  170,  125,   54,  16,   4,  1;
  0, 28, 270,  585,  516,  259,  84,  21,  4, 1;
  0, 51, 729, 2048, 2232, 1296, 480, 128, 27, 5, 1;

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Frank Ruskey, Number of q-ary Lyndon words with given trace mod q
Frank Ruskey, Number of monic irreducible polynomials over GF(q) with given trace
Frank Ruskey, Number of Lyndon words over GF(q) with given trace

Columns include A000048, A046211, A054660, A054662, A054666, A373277, A300674, A300675.

T[n_, k_]:=Sum[Boole[GCD[d, k] == 1]  MoebiusMu[d] k^((n - k + 1)/d), {d, Divisors[n - k + 1]}] /(k(n - k + 1)); Flatten[Table[T[n, k], {n, 12}, {k, n}]] (* Indranil Ghosh, Mar 27 2017 *)

for(n=1, 11, for(k=1, n, print1( sum(d=1,n-k+1, if(Mod(n-k+1, d)==0 && gcd(d, k)==1, moebius(d)*k^((n-k+1)/d), 0)/(k*(n-k+1)) ),", ");); print();) \\ Andrew Howroyd, Mar 26 2017

T(n, k) = (Sum_{d | n-k+1, gcd(d, k)=1} mu(d)*k^((n-k+1)/d))/(k*(n-k+1)).

Name clarified by Andrew Howroyd, Mar 26 2017

A373283 Expansion of Sum_{k>=0} x^(7^k) / (1 - 7*x^(7^k)).

Original entry on oeis.org

1, 7, 49, 343, 2401, 16807, 117650, 823543, 5764801, 40353607, 282475249, 1977326743, 13841287201, 96889010414, 678223072849, 4747561509943, 33232930569601, 232630513987207, 1628413597910449, 11398895185373143, 79792266297612050, 558545864083284007

Offset: 1

Seiichi Manyama, May 30 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A187767, A373279, A373280, A373281, A373282.

Cf. A000420, A373217, A373277.

b(n, k) = sumdiv(n, d, (gcd(d, k)==1)*(moebius(d)*k^(n/d)))/(k*n);
a(n, k=7) = sumdiv(n, d, d*b(d, k));

G.f. A(x) satisfies A(x) = x/(1 - 7*x) + A(x^7).
If n == 0 (mod 7), a(n) = 7^n + a(n/7) otherwise a(n) = 7^n.
a(n) = Sum_{d|n} d * A373277(d).

A386647 G.f. A(x) satisfies: A(x)^7 = A(x^7) / (1 - 7*x).

Original entry on oeis.org

1, 1, 4, 20, 110, 638, 3828, 23515, 146968, 930797, 5957100, 38450370, 249927394, 1634140604, 10738638021, 70875009760, 469546933535, 3121106054760, 20807373517870, 139080864081230, 931841783576460, 6256651942091035, 42090203778813320, 283651372136401905, 1914646755015446620

Offset: 1

Paul D. Hanna, Aug 11 2025

nonn

The EULER transform of A373277, where A373277 is the number of certain monic irreducible polynomials over GF(7).
Compare g.f. to: F(x)^2 = F(x^2)/(1 - 2*x) where F(x) is the g.f. of A123916, the EULER transform of A000048.
Compare g.f. to: G(x)^3 = G(x^3)/(1 - 3*x) where G(x) is the g.f. of A271929, the EULER transform of A046211.
Compare g.f. to: H(x)^5 = H(x^5)/(1 - 5*x) where H(x) is the g.f. of A372535, the EULER transform of A054662.

			G.f.: A(x) = x + x^2 + 4*x^3 + 20*x^4 + 110*x^5 + 638*x^6 + 3828*x^7 + 23515*x^8 + 146968*x^9 + 930797*x^10 + 5957100*x^11 + 38450370*x^12 +...
where A(x)^7 = A(x^7) / (1 - 7*x).
Also, when expressed as the EULER transform of A373277,
A(x) = x/( (1-x) * (1-x^2)^3 * (1-x^3)^16 * (1-x^4)^84 * (1-x^5)^480 * (1-x^6)^2792 * (1-x^7)^16807 * (1-x^8)^102900 * ... * (1-x^n)^A373277(n) * ... ).
RELATED SERIES.
A(x)^7 = x^7 + 7*x^8 + 49*x^9 + 343*x^10 + 2401*x^11 + 16807*x^12 + 117649*x^13 + 823544*x^14 + 5764808*x^15 + ...

Paul D. Hanna, Table of n, a(n) for n = 1..700

Cf. A123916, A271929, A372535, A373277.

{a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^7)/(1 - 7*x +x*O(x^n)))^(1/7)); polcoeff(A, n)}
for(n=1, 50, print1(a(n), ", "))

/* EULER transform of A373277 */
{A373277(n) = 1/(7*n) * sumdiv(n, d, (gcd(d, 7)==1)*(moebius(d)*7^(n/d)))} \\ after Seiichi Manyama in A373277
{a(n) = my(A = x/prod(m=1, n, (1-x^m +x*O(x^n))^A373277(m))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^7 = A(x^7) / (1 - 7*x).
(2) A(x) = x / Product_{n>=1} (1 - x^n)^A373277(n).
a(n) ~ c * 7^n / n^(6/7), where c = 0.02181670654997947129840613123487745678041711647162749305767393184541296... - Vaclav Kotesovec, Aug 12 2025

cp's OEIS Frontend

A110540 Invertible triangle: T(n,k) = number of k-ary Lyndon words of length n-k+1 with trace 1 modulo k.

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A373283 Expansion of Sum_{k>=0} x^(7^k) / (1 - 7*x^(7^k)).

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A386647 G.f. A(x) satisfies: A(x)^7 = A(x^7) / (1 - 7*x).

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